Spatial random permutations and Poisson-Dirichlet law of cycle lengths

TL;DR

This paper investigates spatial permutations with cycle weights, revealing a phase transition at a critical density where long cycles become macroscopic and follow a Poisson-Dirichlet distribution.

Contribution

It establishes the occurrence of a phase transition and characterizes the cycle length distribution as Poisson-Dirichlet in the supercritical regime.

Findings

Existence of a critical density for phase transition

Long cycles become macroscopic beyond critical density

Cycle lengths follow Poisson-Dirichlet law

Abstract

We study spatial permutations with cycle weights that are bounded or slowly diverging. We show that a phase transition occurs at an explicit critical density. The long cycles are macroscopic and their cycle lengths satisfy a Poisson-Dirichlet law.